Approximation of $\zeta(3)$

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What can we add to approximation (or perform) $$\zeta(3)\approx\sqrt{\frac{4}{5}+\frac{\pi^2}{6}-\tanh(2\pi)}$$ for transform it into the equation?

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No closed form is known for Apéry's constant, so there is no known way to convert your approximation into an equation. $$\frac{11888848 }{11888849} \sqrt{\frac{4}{5}+\frac{\pi ^2}{6}-\tanh (2 \pi )}$$ is a slight, boring improvement, reducing the relative error by about $7$ orders of magnitude.